CLIFFORD AND GRASSMANN HOPF ALGEBRAS VIA THE ...

add(add(BW[i,j]*EV(bas[i],y)*EV(bas[j],x),i=1..2^dim_V),j=1..2^dim_V)
end proc:
> Bsw:=proc(x,i) tcollect(gswitch(contract(&gco(&gco(x,i+1),i),i+1,bw),i))
end proc:
> eq3:=Bsw(Bsw(Bsw(eq0,1),2),1):
> eq4:=Bsw(Bsw(Bsw(eq0,2),1),2):
> is(eq3=eq4);
true
which proves our claim in dim_V=2. As a last demonstration, we check the
remaining properties of a quasi triangular structure to be valid for an exponentially
generated endomap where R S is the convolutive inverse of R.
R(S(a),b)=R S (a, b), R S (a, S(b)) = R(a, b), R(S(a),S(b)) = R(a, b).
Hence we generate the endomap ‘R’ and apply to its arguments the Graßmann
antipode:
> R:=’R’:
> out[1]:=‘R‘=matrix(2^dim_V,2^dim_V,(i,j)->
scalarpart(cmul[R](bas[i],bas[j]))):
> out[2]:=‘RS‘=matrix(2^dim_V,2^dim_V,(i,j)->
scalarpart(cmul[R](gantipode(bas[i]),bas[j]))):
> out[1];out[2];
⎡
⎢ 1 0 0 0
⎢ 0 R1, 1 R1, 2 0
R = ⎢ 0 R2, 1 R2, 2 0
⎣
⎤
⎥
⎦
0 0 0 R2, 1 R1, 2 − R2, 2 R1, 1
⎡
⎢ 1 0 0 0
⎢ 0 −R1, 1 −R1, 2 0
RS = ⎢ 0 −R2, 1 −R2, 2 0
⎣
⎤
⎥
⎦
0 0 0 R2, 1 R1, 2 − R2, 2 R1, 1
> out[3]:=‘RSS‘=matrix(2^dim_V,2^dim_V,(i,j)->
scalarpart(cmul[R](gantipode(bas[i]),gantipode(bas[j])))):
> out[4]:=‘SR‘=matrix(2^dim_V,2^dim_V,(i,j)->
scalarpart(cmul[R](bas[i],gantipode(bas[j])))):
> out[3];out[4];
17